Anharmonic motions leading to atomic rearrangements are essential for proteins to function. We report here the first far-infrared absorption studies on anharmonities of low frequency modes in protein model compounds, namely homopolymers of L-phenylalanine, L-Alanine, L-Leeucine, and L-Tryptophan. The bright synchrotron light source was utilized in collecting high signal-to-noise data, so that it is possible to perform quantitative data analysis. The far-infrared absorption spectra of each model compound were collected at two temperatures (100 K and 295 K) in the region of 20-450 cm-1. The results show that (1) each of the homopolymers exhibit a small number of absorption bands in the region of 50 to 150 cm-1, and the peak positions of these bands are well resolved; (2) the peak frequencies of far-infrared bands are temperature dependent, and decrease as temperature is increased from 100 K to 295 K; (3) the band-widths of these low frequency modes, obtained from Gaussian fits are narrower at 100K as compared to those at 295 K. These results strongly infer that the low frequency modes in protein model compounds are highly anharmonic. The results are consistent with various computational studies, as well as indirect experimental studies. This method can be further extended to study anharmonicities of functionally important low frequency models in proteins, including hydrogen bonding interactions, metal-ligand interactions, and collective modes in proteins.